Monte Carlo Simulation of Ising Model and Phase Transition Studies

1. Monte Carlo Simulation of Ising Model and Phase Transition Studies Yu Sun*, Yilin Wu** *Department of Electric Engineering, University of Notre Dame **Department of Physics, University of Notre Dame Instructor: Prof. Mark Alber, Department of Mathematics, University of Notre Dame

2. Outline • Describe the Ising model for magnetism; • Introduce the Monte Carlo simulation method as well as the Metropolis algorithm; • Present our Monte Carlo simulation results for Ising model and discuss its properties, especially the phase transition behavior.

3. Introduction to Magnetism • Magnetic susceptibilityχ: • Types of magnetic materials: • 1. Diamagnetic: χ<0 and constant (Helium); • 2. Paramagnetic: magnetic susceptibility χ>0 and χ∝1/T (Rare earth); • 3. Ferromagnetic: Iron. Below a critical temperature (Curie temperature), χ depends on magnetic field, and the M-H diagram shows a hysteresis loop; above this temperature, the material becomes paramagnetic; • 4. Anti-Ferromagnetic: Below a critical temperature, χ∝T; above this temperature, the material becomes paramagnetic. (MnO) Hysteresis loop

4. Ising Model(2D) • A lattice model proposed to interpret ferromagnetism in materials(1925). • Basic idea: Elementary particles have an intrinsic property called “spin”. Spins carry magnetic moments. The magnetism of a bulk material is made up of the magnetic dipole moments of the atomic spins inside the material. • Ising model postulates a lattice with a spin σ(or magnetic dipole moment) on each site, defining the following Hamiltonian: • E is total energy of the system, J is the nearest spin-spin interaction energy, H is external magnetic field. σ=+1 or -1.

5. Ising Model(2D) • Thermalproperties are defined, and computed, by the partition function, which is the normalization factor of the probability of a thermodynamic state: • Using Z(T), we can calculate the specific heat C, and magnetic susceptibilityχ

6. Phase transitions • The abrupt sudden change in physical properties of the thermodynamic system around some critical value of thermodynamic variables(such astemperature). A particular quantity is the specific heat. • Ehrenfest classification of Phase Transition: • First-order phase transitions exhibit a discontinuity in the first derivative of the chemical potential with a thermodynamic variable. Such as solid/liquid/gas transitions. • Second-order phase transitions(also called continuous phase transition)have a discontinuity or divergence in a second derivative of the chemical potential with thermodynamic variables.

7. Phase transitions • C and χaresecond derivative of chemical potential with T and H separately. • Onsager (1944) obtained the exact solution for 2D Ising model without external field. The solution shows that there exists second order phase transition in C andχ, because they diverge at some critical value of temperature (Tc≈2.269 in unit of (1/Boltzmann constant)). The studies can explain the ferromagnetic to paramagnetic transition of materials. • Monte Carlo simulations also reveal the phase transition properties of Ising model.

8. Monte Carlo method and Metropolis Algorithm • Monte Carlo:A method using pseudorandom number to simulate the random thermal fluctuation from state to state of a system; • The probability of a particular state αfollows Boltzmann distribution: • In theory, sum over all possible states to calculate the statistical mean values of a physical quantity, weighing each state based on its Boltzmann factor; • Metropolis algorithm (importance sampling technique): 1.Flip one randomly picked spin; 2.Calculate the total energy difference between new and old spin state δE=E(new)-E(old); 3. If δE>0, the probability to accept the new state P(old->new) = exp[-δE/kT],otherwiseP(old->new) = 1.

9. Simulation settings • Set the spin-spin interaction energy J=1, Boltzmann constant k=1, Bohr magneton • The unit of Energy is J; the unit of temperature T is

10. Simulation interface

11. Results: Energy per spin versus Temperature (Zero external field). The derivative C=dE/dT diverges at around Tc≈2.269.

12. Results: C versus T. Specific heat divergence is shown more clearly at Tc≈2.269 in this figure. Second order phase transition occurs.

13. Results: Magnetization per spin (Zero external field), T=1.5, 2.0. The figures show spontaneous magnetization (most of the spins align in the same direction).

14. Results: Magnetization per spin (Zero external field), T=2.25, 4.0. Fluctuations become more significant near Tc≈2.269. For T far above Tc, M oscillates around 0.

15. Results: Magnetization per spin versus Temperature (Zero external field).

16. Results: Magnetic susceptibilityχ versus T. χdiverges at around Tc≈2.269. It is second order phase transition. Above Tc, it is paramagnetic.

17. Results: Magnetization per spin versus External field H at T= 0.2. It shows a hysteresis loop, characteristic of ferromagnetic materials.

18. Summary of Results • Demonstrate that second order phase transition of specific heat C and magnetic susceptibilityχoccur at Tc≈2.269, as predicted by Onsager’s exact solution. • Demonstrate the existence of spontaneous magnetization and hysteresis loop below Tc≈2.269 (J>0). These show that the system is ferromagnetic below Tc. • Combing these results, the ferromagnetic to paramagnetic phase transition of 2D Ising model is demonstrated.